The quality and value of faceted gem diamonds are often described in terms of the “four C's”: carat weight, color, clarity, and cut. Weight is the most objective, because it is measured directly on a balance. Color and clarity are factors for which grading standards have been established by GIA, among others. Clamor for the standardization of cut, and calls for a simple cut grading system, have been heard sporadically over the last 27 years, gaining strength recently (Shor, 1993, 1997; Nestlebaum, 1996, 1997). Unlike color and clarity, for which diamond trading, consistent teaching, and laboratory practice have created a general consensus, there are a number of different systems for grading cut in round brilliants. As described in greater detail herein, these systems are based on relatively simple assumptions about the relationship between the proportions and appearance of the round brilliant diamond. Inherent in these systems is the premise that there is one set (or a narrow range) of preferred proportions for round brilliants, and that any deviation from this set of proportions diminishes the attractiveness of a diamond. However, no system described to date has adequately accounted for the rather complex relationship between cut proportions and two of the features within the canonical description of diamond appearance—fire and scintillation.
Diamond manufacturing has undergone considerable change during the past century. For the most part, diamonds have been cut within very close proportion tolerances, both to save weight while maximizing appearance and to account for local market preferences (Caspi, 1997). Differences in proportions can produce noticeable differences in appearance in round-brilliant-cut diamonds. Within this single cutting style, there is substantial debate—and some strongly held views—about which proportions yield the best face-up appearance (Federman, 1997). Yet face-up appearance depends as well on many intrinsic physical and optical properties of diamond as a material, and on the way these properties govern the paths of light through the faceted gemstone. (Other properties particular to each stone, such as polish quality, symmetry, and the presence of inclusions also effect the paths of light through the gemstone).
Diamond appearance is described chiefly in terms of brilliance (white light returned through the crown), fire (the visible extent of light dispersion into spectral colors), and scintillation (flashes of light reflected from the crown). Yet each of these terms cannot be expressed mathematically without making some assumptions and qualifications. Many aspects of diamond evaluation with respect to brilliance are described in “Modeling the Appearance of the Round Brilliant Cut Diamond: An Analysis of Brilliance.” Gems & Gemology, Vol. 34, No. 3, pp. 158–183 (which is hereby incorporated by reference).
Several analyses of the round brilliant cut have been published, starting with Wade (1916). Best known are Tolkowsky's (1919) calculations of the proportions that he believed would optimize the appearance of the round-brilliant-cut diamond. However, Tolkowsky's calculations involved two-dimensional images as graphical and mathematical models. These were used to solve sets of relatively simple equations that described what was considered to be the brilliance of a polished round brilliant diamond. (Tolkowsky did include a simple analysis of fire, but it was not central to his model).
The issues raised by diamond cut are beneficially resolved by considering the complex combination of physical factors that influence the appearance of a faceted diamond (e.g., the interaction of light with diamond as a material, the shape of a given polished diamond, the quality of its surface polish, the type of light source, and the illumination and viewing conditions), and incorporating these into an analysis of that appearance.
Diamond faceting began in about the 1400s and progressed in stages toward the round brilliant we know today (see Tillander, 1966, 1995). In his early mathematical model of the behavior of light in fashioned diamonds, Tolkowsky (1919) used principles from geometric optics to explore how light rays behave in a prism that has a high refractive index. He then applied these results to a two-dimensional model of a round brilliant with a knife-edge girdle, using a single refractive index (that is, only one color of light), and plotted the paths of some illustrative light rays.
Tolkowsky assumed that a light ray is either totally internally reflected or totally refracted out of the diamond, and he calculated the pavilion angle needed to internally reflect a ray of light entering the stone vertically through the table. He followed that ray to the other side of the pavilion and found that a shallower angle is needed there to achieve a second internal reflection. Since it is impossible to create substantially different angles on either side of the pavilion in a symmetrical round brilliant diamond, he next considered a ray that entered the table at a shallow angle. Ultimately, he chose a pavilion angle that permitted this ray to exit through a bezel facet at a high angle, claiming that such an exit direction would allow the dispersion of that ray to be seen clearly. Tolkowsky also used this limiting case of the ray that enters the table at a low angle and exits through the bezel to choose a table size that he claimed would allow the most fire. He concluded by proposing angles and proportions for a round brilliant that he believed best balanced the brilliance and fire of a polished diamond, and then he compared them to some cutting proportions that were typical at that time. However, since Tolkowsky only considered one refractive index, he could not verify the extent to which any of his rays would be dispersed. Nor did he calculate the light loss through the pavilion for rays that enter the diamond at high angles.
Over the next 80 years, other researchers familiar with this work produced their own analyses, with varying results. It is interesting (and somewhat surprising) to realize that despite the numerous possible combinations of proportions for a standard round brilliant, in many cases each researcher arrived at a single set of proportions that he concluded produced an appearance that was superior to all others. Currently, many gem grading laboratories and trade organizations that issue cut grades use narrow ranges of proportions to classify cuts, including what they consider to be best.
Several cut researchers, but not Tolkowsky, used “Ideal” to describe their sets of proportions. Today, in addition to systems that incorporate “Ideal” in their names, many people use this term to refer to measurements similar to Tolkowsky's proportions, but with a somewhat larger table (which, at the same crown angle, yields a smaller crown height percentage). This is what we mean when we use “Ideal” herein.
Numerous standard light modeling programs have also been long available for modeling light refractive objects. E.g., Dadoun, et al., The Geometry of Beam Tracing, ACM Symposium on Computational Geometry, 1985, p. 55–61; Oliver Devillers, Tools to Study the Efficiency of Space Subdivision for Ray Tracing; Proceedings of PixIm '89 Conference; Pub. Gagalowicz, Paris; Heckbert, Beam Tracing Polygonal Objects, Ed. Computer Graphics, SIGGRAPH '84 Proceedings, Vol. 18, No. 3, p. 119–127; Shinya et al., Principles and Applications of Pencil Tracing, SIGGRAPH '87 Proceedings, Vol. 21, No. 4, p. 45–54; Analysis of Algorithm for Fast Ray Tracing Using Uniform Space Subdivision, Journal of Visual Computer, Vol. 4, No. 1, p. 65–83. However, regardless of what standard light modeling technique is used, the diamond modeling programs to date have failed to define effective metrics for diamond cut evaluation. See e.g., (Tognoni, 1990) (Astric et al., 192) (Lawrence, 1998) (Shor 1998). Consequently, there is a need for a computer modeling program that enables a user to make a cut grade using a meaningful diamond analysis metric. Previously, Dodson (1979) used a three-dimensional model of a fully faceted round brilliant diamond to devise metrics for brilliance, fire, and “sparkliness” (scintillation). His mathematical model employed a full sphere of approximately diffuse illumination centered on the diamond's table. His results were presented as graphs of brilliance, fire, and sparkliness for 120 proportion combinations. They show the complex interdependence of all three appearance aspects on pavilion angle, crown height, and table size. However, Dodson simplified his model calculations by tracing rays from few directions and of few colors. He reduced the model output to one-dimensional data by using the reflection-spot technique of Rosch (S. Rosch, 1927, Zeitschrift Kristallographie, Vol. 65, pp. 46–48.), and then spinning that computed pattern and evaluating various aspects of the concentric circles that result. Spinning the data in this way greatly reduces the richness of information, adversely affecting the aptness of the metrics based on it. Thus, there is a need for diamond evaluation that comprises fire and scintillation analysis.